Chandrasekaran, Venkat and Chertkov, Misha and Gamarnik, David and Shah, Devavrat and Shin, Jinwoo (2011) Counting Independent Sets Using the Bethe Approximation. SIAM Journal on Discrete Mathematics, 25 (2). pp. 10121034. ISSN 08954801. https://resolver.caltech.edu/CaltechAUTHORS:20121008092632684

PDF
 Published Version
See Usage Policy. 238Kb 
Use this Persistent URL to link to this item: https://resolver.caltech.edu/CaltechAUTHORS:20121008092632684
Abstract
We consider the #Pcomplete problem of counting the number of independent sets in a given graph. Our interest is in understanding the effectiveness of the popular belief propagation (BP) heuristic. BP is a simple iterative algorithm that is known to have at least one fixed point, where each fixed point corresponds to a stationary point of the Bethe free energy (introduced by Yedidia, Freeman, and Weiss [IEEE Trans. Inform. Theory, 51 (2004), pp. 2282–2312] in recognition of Bethe’s earlier work in 1935). The evaluation of the Bethe free energy at such a stationary point (or BP fixed point) leads to the Bethe approximation for the number of independent sets of the given graph. BP is not known to converge in general, nor is an efficient, convergent procedure for finding stationary points of the Bethe free energy known. Furthermore, the effectiveness of the Bethe approximation is not well understood. As the first result of this paper we propose a BPlike algorithm that always converges to a stationary point of the Bethe free energy for any graph for the independent set problem. This procedure finds an εapproximate stationary point in O(n^2d^42^dε^(4)log^3(nε^(1))) iterations for a graph of n nodes with maxdegree d. We study the quality of the resulting Bethe approximation using the recently developed “loop series” framework of Chertkov and Chernyak [J. Stat. Mech. Theory Exp., 6 (2006), P06009]. As this characterization is applicable only for exact stationary points of the Bethe free energy, we provide a slightly modified characterization that holds for εapproximate stationary points. We establish that for any graph on n nodes with maxdegree d and girth larger than 8d log^2 n, the multiplicative error between the number of independent sets and the Bethe approximation decays as 1+O(n^(−γ)) for some γ>0. This provides a deterministic counting algorithm that leads to strictly different results compared to a recent result of Weitz [in Proceedings of the ThirtyEighth Annual ACM Symposium on Theory of Computing, ACM Press, New York, 2006, pp. 140–149]. Finally, as a consequence of our analysis we prove that the Bethe approximation is exceedingly good for a random 3regular graph conditioned on the shortest cycle cover conjecture of Alon and Tarsi [SIAM J. Algebr. Discrete Methods, 6 (1985), pp. 345–350] being true.
Item Type:  Article  

Related URLs: 
 
Additional Information:  © 2011 Society for Industrial and Applied Mathematics. Received by the editors August 10, 2009; accepted for publication (in revised form) July 16, 2010; published electronically July 1, 2011. This work was supported in part by NSF EMT/MISC collaborative project 0829893.  
Funders: 
 
Subject Keywords:  Bethe free energy, independent set, belief propagation, loop series  
Issue or Number:  2  
Classification Code:  AMS subject classifications: 68Q87, 68W25, 68R10  
Record Number:  CaltechAUTHORS:20121008092632684  
Persistent URL:  https://resolver.caltech.edu/CaltechAUTHORS:20121008092632684  
Official Citation:  Counting Independent Sets Using the Bethe Approximation Venkat Chandrasekaran, Misha Chertkov, David Gamarnik, Devavrat Shah, and Jinwoo Shin SIAM J. Discrete Math. 252 (2011), pp. 10121034http://dx.doi.org/10.1137/090767145  
Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  34742  
Collection:  CaltechAUTHORS  
Deposited By:  Ruth Sustaita  
Deposited On:  08 Oct 2012 16:50  
Last Modified:  03 Oct 2019 04:21 
Repository Staff Only: item control page